Abdeslem Hafid BENTBIB - Academia.edu (original) (raw)

Papers by Abdeslem Hafid BENTBIB

Journal of Computational and Applied Mathematics, Oct 1, 2019

In this paper, we present a new approach for model order reduction in large-scale dynamical syste... more In this paper, we present a new approach for model order reduction in large-scale dynamical systems, with multiple inputs and multiple outputs (MIMO). This approach will be named: Adaptive Block Tangential Arnoldi Algorithm (ABTAA) and is based on interpolation via block tangential Krylov subspaces requiring the selection of shifts and tangent directions via an adaptive procedure. We give some algebraic properties and present some numerical examples to show the effectiveness of the proposed method.

Applicationes Mathematicae, 2016

We propose an adaptive model reduction algorithm for computing a reduced-order model of dynamical... more We propose an adaptive model reduction algorithm for computing a reduced-order model of dynamical multi-input and multi-output (MIMO) linear time independent (LTI) dynamical systems. The process is based on multipoint moment matching. Moreover, we develop new simple Lanczos-like equations for the rational block case, and we use them to derive simple residual error expressions. An adaptive method for choosing the interpolation points is also proposed. Finally, some numerical experiments are reported to show the effectiveness of the new adaptive modified rational block Lanczos (AMRBL) process when applied to stable linear LTI dynamical systems.

Numerical Algorithms, Sep 12, 2022

Electronic Transactions on Numerical Analysis, 2019

The need to compute the trace of a large matrix that is not explicitly known, such as the matrix ... more The need to compute the trace of a large matrix that is not explicitly known, such as the matrix exp(A), where A is a large symmetric matrix, arises in various applications including in network analysis. The global Lanczos method is a block method that can be applied to compute an approximation of the trace. When the block size is one, this method simplifies to the standard Lanczos method. It is known that for some matrix functions and matrices, the extended Lanczos method, which uses subspaces with both positive and negative powers of A, can give faster convergence than the standard Lanczos method, which uses subspaces with nonnegative powers of A only. This suggests that it may be beneficial to use an extended global Lanczos method instead of the (standard) global Lanczos method. This paper describes an extended global Lanczos method and discusses properties of the associated Gauss-Laurent quadrature rules. Computed examples that illustrate the performance of the extended global Lanczos method are presented.

Journal of Computational and Applied Mathematics, Aug 1, 2020

This paper presents an efficient algorithm to solve total variation (TV) regularizations of image... more This paper presents an efficient algorithm to solve total variation (TV) regularizations of images contaminated by a both blur and noise. The unconstrained structure of the problem suggests that one can solve a constrained optimization problem by transforming the original unconstrained minimization problem to an equivalent constrained minimization one. An augmented Lagrangian method is developed to handle the constraints when the model is given with matrix variables, and an alternating direction method (ADM) is used to iteratively find solutions. The solutions of some sub-problems are belonging to subspaces generated by application of successive orthogonal projections onto a class of generalized matrix Krylov subspaces of increasing dimension.

Eurasian journal of mathematical and computer applications, 2023

In this paper, we study the ill-posed problem using total variation regularization. To solve such... more In this paper, we study the ill-posed problem using total variation regularization. To solve such a problem, we use an alternating direction method of multipliers to split our problem into two sub-problems. The novelty of our paper is in the use of the conditional gradient total variation method (CGTV) we have recently introduced. The second split- ting sub-problem is solved by transforming the obtained optimization problem to a general Sylvester matrix equation and then an orthogonal projection method is used to solve the obtained matrix equation. We give proof of the convergence of this method. Some numerical examples and applications to image restoration are given to illustrate the effectiveness of the proposed method.

Electronic Transactions on Numerical Analysis, 2019

Méthodes de sous espaces de Krylov préconditionnées pour les problèmes de pointselle avec plusieu... more Méthodes de sous espaces de Krylov préconditionnées pour les problèmes de pointselle avec plusieurs seconds membres Résumé La résolution numérique des problèmes de point-selle a eu une attention particulière ces dernières années. A titre d'exemple, la mécanique des fluides et solides conduit souvent à des problèmes de point-selle. Ces problèmes se présentent généralement par des équations aux dérivées partielles que nous linéarisons et discrétisons. Le problème linéaire obtenu est souvent mal conditionné. Le résoudre par des méthodes itératives standard n'est donc pas approprié. En plus, lorsque la taille du problème est grande, il est nécessaire de procéder par des méthodes de projections. Nous nous intéressons dans ce sujet de thèse à développer des méthodes numériques robustes et efficaces de résolution numérique de problèmes de point-selle. Nous appliquons les méthodes de Krylov avec des techniques de préconditionnement bien adaptées à la résolution de problèmes de point selle. L'efficacité de ces méthodes à été montrée dans les tests numériques. Mots clés : point-selle, préconditionnement, krylov, produit de kronecker, produit de diamant Preconditioned global Krylov subspace methods for solving saddle point problems with multiple right-hand sides

arXiv (Cornell University), Feb 20, 2021

In the present paper we propose two new algorithms of tensor completion for threeorder tensors. T... more In the present paper we propose two new algorithms of tensor completion for threeorder tensors. The proposed methods consist in minimizing the average rank of the underlying tensor using its approximate function namely the tensor nuclear norm and then the recovered data will be obtained by using the total variation regularisation technique. We will adopt the Alternating Direction Method of Multipliers (ADM), using the tensor T-product, to solve the main optimization problems associated to the two algorithms. In the last section, we present some numerical experiments and comparisons with the most known image completion methods.

arXiv (Cornell University), Mar 31, 2020

The numerical computation of matrix functions such as f (A)V , where A is an n × n large and spar... more The numerical computation of matrix functions such as f (A)V , where A is an n × n large and sparse square matrix, V is an n × p block with p ≪ n and f is a nonlinear matrix function, arises in various applications such as network analysis (f (t) = exp(t) or f (t) = t 3), machine learning (f (t) = log(t)), theory of quantum chromodynamics (f (t) = t 1/2), electronic structure computation, and others. In this work, we propose the use of global extended-rational Arnoldi method for computing approximations of such expressions. The derived method projects the initial problem onto an global extended-rational Krylov subspace RK e m (A,V) = span({ m ∏ i=1 (A − s i I n) −1 V,. . ., (A − s 1 I n) −1 V,V , AV,. . ., A m−1 V }) of a low dimension. An adaptive procedure for the selection of shift parameters {s 1 ,. .. , s m } is given. The proposed method is also applied to solve parameter dependent systems. Numerical examples are presented to show the performance of the global extended-rational Arnoldi for these problems.

Electronic Transactions on Numerical Analysis, 2021

In this paper, we are interested in finding an approximate solutionX of the tensor least-squares ... more In this paper, we are interested in finding an approximate solutionX of the tensor least-squares minimization problem min X X × 1 A (1) × 2 A (2) × 3 • • • × N A (N) − G , where G ∈ R J 1 ×J 2 ו••×J N and A (i) ∈ R J i ×I i (i = 1,. .. , N) are known and X ∈ R I 1 ×I 2 ו••×I N is the unknown tensor to be approximated. Our approach is based on two steps. Firstly, we apply the CP or HOSVD decomposition to the right-hand side tensor G. Secondly, we perform the well-known Golub-Kahan bidiagonalization for each coefficient matrix A (i) (i = 1,. .. , N) to obtain a reduced tensor least-squares minimization problem. This type of equations may appear in color image and video restorations as we described below. Some numerical tests are performed to show the effectiveness of our proposed method.

Bit Numerical Mathematics, Feb 3, 2022

Bit Numerical Mathematics, May 4, 2018

The original version of this article unfortunately contained a mistake. The presentation of Algor... more The original version of this article unfortunately contained a mistake. The presentation of Algorithm 4 was incorrect in this article. The corrected Algorithm 4 is given below.

Calcolo, Sep 16, 2016

In the present paper, we propose a new method to inexpensively determine a suitable value of the ... more In the present paper, we propose a new method to inexpensively determine a suitable value of the regularization parameter and an associated approximate solution, when solving ill-conditioned linear system of equations with multiple righthand sides contaminated by errors. The proposed method is based on the symmetric block Lanczos algorithm, in connection with block Gauss quadrature rules to inexpensively approximate matrix-valued function of the form W T f (A)W , where W ∈ R n×k , k n, and A ∈ R n×n is a symmetric matrix. Keywords Block Lanczos • Gauss quadrature • Ill-posed • Regularization • Tikhonov Mathematics Subject Classification 65F • 15A 1 Introduction Many applications require the solution of several ill-conditioning systems of equations with a right hand side contaminated by an additive error,

Journal of Scientific Computing, Nov 14, 2017

In this paper, we present a new approach for model order reduction problems, with multiple inputs... more In this paper, we present a new approach for model order reduction problems, with multiple inputs and multiple outputs, named: Adaptive Global Tangential Arnoldi Algorithm. This method is based on a generalization of the global Arnoldi algorithm. The selection of the shifts and the tangent directions are done with an adaptive procedure. We give some algebraic properties and present some numerical examples to show the effectiveness of the proposed algorithm.

Mathematics, Mar 27, 2017

In the present paper, we consider the large scale Stein matrix equation with a low-rank constant ... more In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term AXB − X + EF T = 0. These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments.

Computers & mathematics with applications, Nov 1, 2015

In the present paper, we consider large scale nonsymmetric matrix Riccati equations with low rank... more In the present paper, we consider large scale nonsymmetric matrix Riccati equations with low rank right hand sides. These matrix equations appear in many applications such as transport theory, Wiener-Hopf factorization of Markov chains, applied probability and others. We show how to apply directly Krylov methods such as the extended block Arnoldi algorithm to get low rank approximate solutions. We also combine the Newton method and block Krylov subspace methods to get approximations of the desired minimal nonnegative solution. We give some theoretical results and report some numerical experiments for the well known transport equation.

Journal of Optimization Theory and Applications

Journal of Scientific Computing

Optimization Methods and Software

Calcolo, 2021

This paper describes solution methods for linear discrete ill-posed problems defined by third ord... more This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in [M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641-658]. A t-product Arnoldi (t-Arnoldi) process is defined and applied to reduce a large-scale Tikhonov regularization problem for third order tensors to a problem of small size. The data may be represented by a laterally oriented matrix or a third order tensor, and the regularization operator is a third order tensor. The discrepancy principle is used to determine the regularization parameter and the number of steps of the t-Arnoldi process. Numerical examples compare results for several solution methods, and illustrate the potential superiority of solution methods that tensorize over solution methods that matricize linear discrete ill-posed problems for third order tensors.

Journal of Computational and Applied Mathematics, Oct 1, 2019

In this paper, we present a new approach for model order reduction in large-scale dynamical syste... more In this paper, we present a new approach for model order reduction in large-scale dynamical systems, with multiple inputs and multiple outputs (MIMO). This approach will be named: Adaptive Block Tangential Arnoldi Algorithm (ABTAA) and is based on interpolation via block tangential Krylov subspaces requiring the selection of shifts and tangent directions via an adaptive procedure. We give some algebraic properties and present some numerical examples to show the effectiveness of the proposed method.

Applicationes Mathematicae, 2016

We propose an adaptive model reduction algorithm for computing a reduced-order model of dynamical... more We propose an adaptive model reduction algorithm for computing a reduced-order model of dynamical multi-input and multi-output (MIMO) linear time independent (LTI) dynamical systems. The process is based on multipoint moment matching. Moreover, we develop new simple Lanczos-like equations for the rational block case, and we use them to derive simple residual error expressions. An adaptive method for choosing the interpolation points is also proposed. Finally, some numerical experiments are reported to show the effectiveness of the new adaptive modified rational block Lanczos (AMRBL) process when applied to stable linear LTI dynamical systems.

Numerical Algorithms, Sep 12, 2022

Electronic Transactions on Numerical Analysis, 2019

The need to compute the trace of a large matrix that is not explicitly known, such as the matrix ... more The need to compute the trace of a large matrix that is not explicitly known, such as the matrix exp(A), where A is a large symmetric matrix, arises in various applications including in network analysis. The global Lanczos method is a block method that can be applied to compute an approximation of the trace. When the block size is one, this method simplifies to the standard Lanczos method. It is known that for some matrix functions and matrices, the extended Lanczos method, which uses subspaces with both positive and negative powers of A, can give faster convergence than the standard Lanczos method, which uses subspaces with nonnegative powers of A only. This suggests that it may be beneficial to use an extended global Lanczos method instead of the (standard) global Lanczos method. This paper describes an extended global Lanczos method and discusses properties of the associated Gauss-Laurent quadrature rules. Computed examples that illustrate the performance of the extended global Lanczos method are presented.

Journal of Computational and Applied Mathematics, Aug 1, 2020

This paper presents an efficient algorithm to solve total variation (TV) regularizations of image... more This paper presents an efficient algorithm to solve total variation (TV) regularizations of images contaminated by a both blur and noise. The unconstrained structure of the problem suggests that one can solve a constrained optimization problem by transforming the original unconstrained minimization problem to an equivalent constrained minimization one. An augmented Lagrangian method is developed to handle the constraints when the model is given with matrix variables, and an alternating direction method (ADM) is used to iteratively find solutions. The solutions of some sub-problems are belonging to subspaces generated by application of successive orthogonal projections onto a class of generalized matrix Krylov subspaces of increasing dimension.

Eurasian journal of mathematical and computer applications, 2023

In this paper, we study the ill-posed problem using total variation regularization. To solve such... more In this paper, we study the ill-posed problem using total variation regularization. To solve such a problem, we use an alternating direction method of multipliers to split our problem into two sub-problems. The novelty of our paper is in the use of the conditional gradient total variation method (CGTV) we have recently introduced. The second split- ting sub-problem is solved by transforming the obtained optimization problem to a general Sylvester matrix equation and then an orthogonal projection method is used to solve the obtained matrix equation. We give proof of the convergence of this method. Some numerical examples and applications to image restoration are given to illustrate the effectiveness of the proposed method.

Electronic Transactions on Numerical Analysis, 2019

Méthodes de sous espaces de Krylov préconditionnées pour les problèmes de pointselle avec plusieu... more Méthodes de sous espaces de Krylov préconditionnées pour les problèmes de pointselle avec plusieurs seconds membres Résumé La résolution numérique des problèmes de point-selle a eu une attention particulière ces dernières années. A titre d'exemple, la mécanique des fluides et solides conduit souvent à des problèmes de point-selle. Ces problèmes se présentent généralement par des équations aux dérivées partielles que nous linéarisons et discrétisons. Le problème linéaire obtenu est souvent mal conditionné. Le résoudre par des méthodes itératives standard n'est donc pas approprié. En plus, lorsque la taille du problème est grande, il est nécessaire de procéder par des méthodes de projections. Nous nous intéressons dans ce sujet de thèse à développer des méthodes numériques robustes et efficaces de résolution numérique de problèmes de point-selle. Nous appliquons les méthodes de Krylov avec des techniques de préconditionnement bien adaptées à la résolution de problèmes de point selle. L'efficacité de ces méthodes à été montrée dans les tests numériques. Mots clés : point-selle, préconditionnement, krylov, produit de kronecker, produit de diamant Preconditioned global Krylov subspace methods for solving saddle point problems with multiple right-hand sides

arXiv (Cornell University), Feb 20, 2021

In the present paper we propose two new algorithms of tensor completion for threeorder tensors. T... more In the present paper we propose two new algorithms of tensor completion for threeorder tensors. The proposed methods consist in minimizing the average rank of the underlying tensor using its approximate function namely the tensor nuclear norm and then the recovered data will be obtained by using the total variation regularisation technique. We will adopt the Alternating Direction Method of Multipliers (ADM), using the tensor T-product, to solve the main optimization problems associated to the two algorithms. In the last section, we present some numerical experiments and comparisons with the most known image completion methods.

arXiv (Cornell University), Mar 31, 2020

The numerical computation of matrix functions such as f (A)V , where A is an n × n large and spar... more The numerical computation of matrix functions such as f (A)V , where A is an n × n large and sparse square matrix, V is an n × p block with p ≪ n and f is a nonlinear matrix function, arises in various applications such as network analysis (f (t) = exp(t) or f (t) = t 3), machine learning (f (t) = log(t)), theory of quantum chromodynamics (f (t) = t 1/2), electronic structure computation, and others. In this work, we propose the use of global extended-rational Arnoldi method for computing approximations of such expressions. The derived method projects the initial problem onto an global extended-rational Krylov subspace RK e m (A,V) = span({ m ∏ i=1 (A − s i I n) −1 V,. . ., (A − s 1 I n) −1 V,V , AV,. . ., A m−1 V }) of a low dimension. An adaptive procedure for the selection of shift parameters {s 1 ,. .. , s m } is given. The proposed method is also applied to solve parameter dependent systems. Numerical examples are presented to show the performance of the global extended-rational Arnoldi for these problems.

Electronic Transactions on Numerical Analysis, 2021

In this paper, we are interested in finding an approximate solutionX of the tensor least-squares ... more In this paper, we are interested in finding an approximate solutionX of the tensor least-squares minimization problem min X X × 1 A (1) × 2 A (2) × 3 • • • × N A (N) − G , where G ∈ R J 1 ×J 2 ו••×J N and A (i) ∈ R J i ×I i (i = 1,. .. , N) are known and X ∈ R I 1 ×I 2 ו••×I N is the unknown tensor to be approximated. Our approach is based on two steps. Firstly, we apply the CP or HOSVD decomposition to the right-hand side tensor G. Secondly, we perform the well-known Golub-Kahan bidiagonalization for each coefficient matrix A (i) (i = 1,. .. , N) to obtain a reduced tensor least-squares minimization problem. This type of equations may appear in color image and video restorations as we described below. Some numerical tests are performed to show the effectiveness of our proposed method.

Bit Numerical Mathematics, Feb 3, 2022

Bit Numerical Mathematics, May 4, 2018

The original version of this article unfortunately contained a mistake. The presentation of Algor... more The original version of this article unfortunately contained a mistake. The presentation of Algorithm 4 was incorrect in this article. The corrected Algorithm 4 is given below.

Calcolo, Sep 16, 2016

In the present paper, we propose a new method to inexpensively determine a suitable value of the ... more In the present paper, we propose a new method to inexpensively determine a suitable value of the regularization parameter and an associated approximate solution, when solving ill-conditioned linear system of equations with multiple righthand sides contaminated by errors. The proposed method is based on the symmetric block Lanczos algorithm, in connection with block Gauss quadrature rules to inexpensively approximate matrix-valued function of the form W T f (A)W , where W ∈ R n×k , k n, and A ∈ R n×n is a symmetric matrix. Keywords Block Lanczos • Gauss quadrature • Ill-posed • Regularization • Tikhonov Mathematics Subject Classification 65F • 15A 1 Introduction Many applications require the solution of several ill-conditioning systems of equations with a right hand side contaminated by an additive error,

Journal of Scientific Computing, Nov 14, 2017

In this paper, we present a new approach for model order reduction problems, with multiple inputs... more In this paper, we present a new approach for model order reduction problems, with multiple inputs and multiple outputs, named: Adaptive Global Tangential Arnoldi Algorithm. This method is based on a generalization of the global Arnoldi algorithm. The selection of the shifts and the tangent directions are done with an adaptive procedure. We give some algebraic properties and present some numerical examples to show the effectiveness of the proposed algorithm.

Mathematics, Mar 27, 2017

In the present paper, we consider the large scale Stein matrix equation with a low-rank constant ... more In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term AXB − X + EF T = 0. These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments.

Computers & mathematics with applications, Nov 1, 2015

In the present paper, we consider large scale nonsymmetric matrix Riccati equations with low rank... more In the present paper, we consider large scale nonsymmetric matrix Riccati equations with low rank right hand sides. These matrix equations appear in many applications such as transport theory, Wiener-Hopf factorization of Markov chains, applied probability and others. We show how to apply directly Krylov methods such as the extended block Arnoldi algorithm to get low rank approximate solutions. We also combine the Newton method and block Krylov subspace methods to get approximations of the desired minimal nonnegative solution. We give some theoretical results and report some numerical experiments for the well known transport equation.

Journal of Optimization Theory and Applications

Journal of Scientific Computing

Optimization Methods and Software

Calcolo, 2021

This paper describes solution methods for linear discrete ill-posed problems defined by third ord... more This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in [M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641-658]. A t-product Arnoldi (t-Arnoldi) process is defined and applied to reduce a large-scale Tikhonov regularization problem for third order tensors to a problem of small size. The data may be represented by a laterally oriented matrix or a third order tensor, and the regularization operator is a third order tensor. The discrepancy principle is used to determine the regularization parameter and the number of steps of the t-Arnoldi process. Numerical examples compare results for several solution methods, and illustrate the potential superiority of solution methods that tensorize over solution methods that matricize linear discrete ill-posed problems for third order tensors.